AFM
Unit 4 Review
Logarithmic & Power Functions
Name______________________________________________
Show work.
1. A pressure altimeter is an instrument that finds the height of an airplane above sea level based on the pressure
𝑝
of the surrounding air. The height h and pressure p are related by the equation ℎ = −26,400 ln (2120) where h
is measured in feet and p is measured in pounds per square foot.
a. Estimate the height above sea level when the pressure is 563 psf.
b. Find the air pressure if the altitude is 20,000 feet above sea level.
2. Noted astronomer Johannes Kepler (1571 – 1630) developed three laws of planetary motion that are used to
this day. Kepler’s Third Law states the orbital period (the time required for one full revolution around the
Sun), in days, 𝑇 for each planet is modeled by the function 𝑇 = 0.2 𝑎1.5, where 𝑎 is the planet’s average
distance from the sun in gigameters (a gigameter is 1 million kilometers). Use the model to predict the orbital
period for Neptune, which is 4997 Gm from the sun on average.
3. When a musician plays an instrument in a concert hall, the sound reverberates off the walls of the hall even
after the musician has stopped playing. The time it takes for the sound to fade away is called the reverberation
time. At the Royal Albert Hall in London, England, the approximate reverberation time T (in seconds) is given
by the function 𝑇 = 5.24 − 0.408 ln 𝑓 where f is the frequency of the sound in hertz (Hz). Suppose a violinist
plays a middle C in the hall. Estimate the reverberation time. Use 261.6 Hz as the frequency of middle C.
4. Water is leaking out of a container with a hole in the bottom. Torricelli’s Law states that at any instant, the
velocity v, in feet per second, with which the water escapes the container, is a power function of d, the depth of
the water, in feet, at that moment. Given 𝑣 = 8(𝑑)0.5, find the velocity of the escaping water when the depth is
2 feet.
5. The Richter scale is used for measuring the magnitude of an earthquake. The Richter magnitude is given by 𝑅 =
0.67 log(0.37𝐸) + 1.46 where E is the energy (in kilowatt-hours) released by the earthquake.
a. An earthquake releases 15,500,000,000 kilowatt-hours of energy. What is the earthquake’s magnitude?
b. How many kilowatt-hours of energy would an earthquake have to release in order to be an 8.5 on the
Richter scale?
6. For mammals and other warm-blooded animals to stay warm requires quire a bit of energy. Temperature loss
is related to surface area, which is related to body weight, and temperature gain is related to circulation, which
is related to pulse rate. In the final analysis, scientists have concluded that the pulse rate 𝑟, in beats per minute,
of mammals can be modeled by the function 𝑟 = 231𝑤 −0.3 , where 𝑤 is body weight in kilograms.
a. Predict the pulse rate for a 450 kg horse.
b. Find the weight of a dog whose pulse rate is 85 beats per minute.